Simplify; express your answer in exponential form. Assume $x\neq 0, y\neq 0$. $\dfrac{{x^{4}}}{{(x^{3}y^{2})^{-5}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${x^{4}}$ to the exponent ${1}$ . Now ${4 \times 1 = 4}$ , so ${x^{4} = x^{4}}$ In the denominator, we can use the distributive property of exponents. ${(x^{3}y^{2})^{-5} = (x^{3})^{-5}(y^{2})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{x^{4}}}{{(x^{3}y^{2})^{-5}}} = \dfrac{{x^{4}}}{{x^{-15}y^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{4}}}{{x^{-15}y^{-10}}} = \dfrac{{x^{4}}}{{x^{-15}}} \cdot \dfrac{{1}}{{y^{-10}}} = x^{{4} - {(-15)}} \cdot y^{- {(-10)}} = x^{19}y^{10}$.